Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-n^3 + 2n^2 + 3n}{7n^3 - 42n^2 + 63n}$
First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-n(n^2 - 2n - 3)} {7n(n^2 - 6n + 9)} $ $ p = -\dfrac{n}{7n} \cdot \dfrac{n^2 - 2n - 3}{n^2 - 6n + 9} $ Simplify: $ p = - \dfrac{1}{7} \cdot \dfrac{n^2 - 2n - 3}{n^2 - 6n + 9}$ Since we are dividing by $n$ , we must remember that $n \neq 0$ Next factor the numerator and denominator. $ p = - \dfrac{1}{7} \cdot \dfrac{(n - 3)(n + 1)}{(n - 3)(n - 3)}$ Assuming $n \neq 3$ , we can cancel the $n - 3$ $ p = - \dfrac{1}{7} \cdot \dfrac{n + 1}{n - 3}$ Therefore: $ p = \dfrac{ -n - 1 }{ 7(n - 3)}$, $n \neq 3$, $n \neq 0$